Intereses nacionales Matriz de Intereses Nacionales (MIN)

Our goal is to examine last survivor annuity prices and the implied market prices of joint-life longevity risk. Base rates of mortality should in principle be related to the mortality experience of annuitants. The currently used annuity life tables in the US and UK annuity market are used as base tables.

A life table gives mortality rates at each age for an individual. It specifies the distribution of the future lifetime random variable Tx for males or females at any

age x, regardless of their marital status. It is the aggregate mortality rates for an individual in the status of being married, single, divorced, or widowed with any period of time after bereavement.

From the semi-Markov joint-life model, without mortality projection at this stage, we can derive marginal mortality rates for the married and the widowed. Assume mortality rates for the single or divorced are the same as the aggregate

rates. Let hx be the percentage of population in the married status, and gx be the

proportion in the widowed status; 1 − hx − gx is for the others. We assume that

the aggregate mortality rate is approximately represented by the equation µaggregate_x = hx

hx+ gx

µmarried_x + gx hx+ gx

µwidowed_x .

Borrowing information from relevant census study on age-specific marital status, we can approximately estimate the values of the Gompertz parameters for the base mortality rates in the semi-Markov joint-life longevity risk model, by equating the approximately mixed single-life mortality rates to the mortality rates in the referred life table for both males and females. Meanwhile, we can also estimate the values of the Gompertz parameters, by directly fitting Gompertz’ law to the mortality rates in the referred life table for both males and females, that is, without the semi- Markov model. The resulting estimates are for individual single-life Gompertz mortality model or independent joint-life mortality rates.

US insurance companies generally use the Annuity 2000 Basic Mortality Table (A2000, for short) as the base mortality for annuity pricing. Using the marital status of the population in 2000 studied by Kreider and Simmons (2003), and the individual mortality rates in A2000, we estimate the parameters for the base rates of mortality in the semi-Markov joint-life longevity model, by fitting the mixed marginal mortality distribution from the semi-Markov joint-life mortality model to the mortality rates in the A2000 life table. The fitting approach is based on a least squares minimization.

Table 4.1 summarizes the estimated parameters for the semi-Markov joint-life mortality model and individual single-life Gompertz mortality model. These pa- rameters are based on the mortality rates at ages beyond 59 in A2000 life table. Single-life Gompertz models specify the aggregate mortality rates for individuals in various marital status. They can be used for independent joint lives or single lives. The parameters displayed in Table 4.1 indicates several important results. Firstly,

The semi-Markov model Single-life model

Parameters Values Parameters Values

γf _{91.9541 γ}f, Inde _90.4699 ξf _{0.1096 ξ}f, Inde _0.1138 γm _{89.1318 γ}m, Inde _87.1418 ξm _{0.0870 ξ}m, Inde _0.0972 af _3.7804 kf _0.3901 am _10.4253 km _0.7754

Table 4.1: Parameter values for base mortality in the semi-Markov joint-life longevity model and individual single-life mortality model, for the US.

the force of mortality in the married status is generally lower than the marginal, or independent, force of mortality of the same age, which represents the combined rate of mortality of the married and the widowed. Secondly, the effect of bereavement will increase the force of mortality after bereavement by a higher level for males than for females, however males recover from bereavement faster than females. This estimated result is consistent to the result in Chapter 2.

Similarly, using the population marital status information provided by the UK Government Actuary’s Department and the UK CMI Series 00 Immediate Annuity Life tables, we can estimate the parameters for the base mortality rates in the semi- Markov joint-life longevity mortality model and single-life Gompertz model applied to the UK annuitants. Table 4.2 summarizes the estimated parameters, which are based on the mortality rates at ages beyond 59 in the CMI Series 00 Immediate Annuity Life tables.

The semi-Markov model Single-life model

Parameters Values Parameters Values

γf _{92.8059 γ}f, Inde _90.0127 ξf _{0.1296 ξ}f, Inde _0.1383 γm _{88.7920 γ}m, Inde _86.6564 ξm _{0.1044 ξ}m, Inde _0.1202 af _8.4790 kf _0.3921 am _8.6868 km _0.3603

Table 4.2: Parameter values for base mortality in the semi-Markov joint-life longevity model and individual single-life mortality model, for the UK.

The estimated parameter values for the UK are slightly different from the values fitted for the US. The difference lies in the parameters for the semi-Markov property, that is, the selection effect of bereavement. From the values for the UK, males and females are subject to a nearly same broken heart effect shortly after bereavement, and they recover from bereavement at a similar speed. Here, we just state the data fitting results. The reasons underlying this difference between the US and UK are beyond the scope of our study.